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a89d282 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 | import OAI.Probability.InvariantIsing.Haar.HaarPolynomialHeat
namespace OAI
/-! Differentiating linear observables of the finite polynomial heat flow. -/
noncomputable section
open Matrix MvPolynomial
namespace InvariantIsing
def haarPolynomialLinearCoordinates {N d : ℕ} (F : MatrixPolynomial N →ₗ[ℝ] ℝ) :
HaarPolynomialCoordinates N d →L[ℝ] ℝ :=
(F.comp ((haarPolynomialSpace N d).subtype.comp
(haarPolynomialCoordinates N d).symm.toLinearMap)).toContinuousLinearMap
lemma haarPolynomialLinearCoordinates_apply {N d : ℕ}
(F : MatrixPolynomial N →ₗ[ℝ] ℝ) (p : haarPolynomialSpace N d) :
haarPolynomialLinearCoordinates F (haarPolynomialCoordinates N d p) = F (p : MatrixPolynomial N) := by
change F (((haarPolynomialCoordinates N d).symm (haarPolynomialCoordinates N d p) :
haarPolynomialSpace N d) : MatrixPolynomial N) = _
rw [LinearEquiv.symm_apply_apply]
lemma haarPolynomialHeat_linear_hasDerivAt {N d : ℕ}
(F : MatrixPolynomial N →ₗ[ℝ] ℝ) (p : haarPolynomialSpace N d) (t : ℝ) :
HasDerivAt (fun s => F ((haarPolynomialHeat N d s p : haarPolynomialSpace N d) : MatrixPolynomial N))
(F (haarPolynomialLaplacian N
((haarPolynomialHeat N d t p : haarPolynomialSpace N d) : MatrixPolynomial N))) t := by
let L := haarPolynomialCoordinateLaplacian N d
let v := haarPolynomialCoordinates N d p
let ev := (haarPolynomialLinearCoordinates F).comp
(ContinuousLinearMap.apply ℝ (HaarPolynomialCoordinates N d) v)
have h := ev.hasFDerivAt.comp_hasDerivAt t (hasDerivAt_exp_smul_const' L t)
change HasDerivAt (fun s => ev (NormedSpace.exp (s • L)))
(ev (L*NormedSpace.exp (t • L))) t at h
have he (s : ℝ) : ev (NormedSpace.exp (s • L)) =
F ((haarPolynomialHeat N d s p : haarPolynomialSpace N d) : MatrixPolynomial N) := rfl
simp_rw [he] at h
convert h using 1
change F _ = haarPolynomialLinearCoordinates F (L (NormedSpace.exp (t • L) v))
rw [← haarPolynomialCoordinates_heat,haarPolynomialCoordinateLaplacian_apply]
exact (haarPolynomialLinearCoordinates_apply F (haarPolynomialRestrictedLaplacian N d
(haarPolynomialHeat N d t p))).symm
end InvariantIsing
end
end OAI
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